Video: AQA GCSE Mathematics Higher Tier Pack 2 • Paper 1 • Question 13

Solve the pair of equations three 𝑥 minus two 𝑦 equals 9.5, 𝑥 minus 𝑦 equals two.


Video Transcript

Solve the pair of equations three 𝑥 minus two 𝑦 equals 9.5, 𝑥 minus 𝑦 equals two.

So what we have here are a pair of simultaneous equations. And to solve them, we need to find 𝑥 and 𝑦 values that will solve both of our equations. We’re looking for one 𝑥 and one 𝑦 value because both of our equations are linear equations.

So the first thing I do with any of these problems is I label our equations. So we’ve got equation one and equation two. I do this so it’s easier to explain what we do as we move through.

In order to solve our equations, there are a couple of methods we could use: the substitution or the elimination method. I’m gonna start by demonstrating the elimination method, and then I’ll move on to the substitution method. So I’m gonna use the elimination method first, and then I’ll show you the substitution method.

So for the elimination method, what we want is the coefficient of our 𝑥 terms or the coefficient of our 𝑦 terms to be the same. But these equations they’re not, cause we have coefficients of 𝑥 of three and one. When we haven’t got a coefficient, that means the coefficient is one. And coefficients of 𝑦 are negative two and negative one.

So what we need to do is actually multiply one of our equations to enable us to have the same coefficient of 𝑥 or 𝑦. So what we could’ve done is multiplied one of our equations, so the second equation, by three to give us the same coefficient of 𝑥. But what I’m going to do is multiply it by two to give us the same coefficient of 𝑦.

And when we do that, we get two 𝑥 minus two 𝑦 equals four. And that’s because two multiplied by 𝑥 is two 𝑥, two multiplied by minus 𝑦 is minus two 𝑦, and two multiplied by two is four. The reason we’re gonna do this is because what we’re doing is we’re multiplying the left-hand side of the equation and the right-hand side of the equation by two. So it stays balanced.

So the next stage in elimination is to eliminate one of our variables. And we can do that because, in equation one and equation three, our variable 𝑦 has the same coefficient, which is negative two. And to do that, what we need to do is either add or subtract the equations from each other.

Well, we know that we’re going to subtract them because our coefficients of 𝑦 have the same sign. And we say that SSS, same sign subtract. If we have different signs, so DSA, then we do different sign and add. So we‘d add the equations together.

It’s worth noting at this point though be careful of a common mistake. If we see in orange I’ve drawn two other equations, so another pair of equations. Here the coefficients that are the same are the 𝑥 coefficients cause they’re both two. However, we can see that the middle signs, so the signs between the 𝑥 and the 𝑦 values, are positive and negative, so they’re different. So often students will go, “Ah, different sign add,” and they’ll add the equations together. However, this is incorrect because the coefficients of the terms are the same, and that’s the coefficients of 𝑥 which are both two. These are in fact positive, both positive, so they would be same sign, so we’d use same sign subtract.

So now what I’m gonna do to eliminate our 𝑦s is do equation one minus equation three. So if we have three 𝑥 minus two 𝑥, this is just gonna be 𝑥. And then negative two 𝑦 minus negative two 𝑦 is gonna be negative two 𝑦 plus two 𝑦. That’s because minus and negative turns it into a plus, so negative two 𝑦 plus two 𝑦 is zero. So we’ve eliminated our 𝑦s as we wanted. And then 𝑥 is gonna be equal to 5.5. And that’s because 9.5 minus four is 5.5.

Okay, great, we’ve found our 𝑥 value. So now what we need to do to find our 𝑦 value is substitute our 𝑥 value, so 𝑥 equals 5.5, into one of our three equations. It doesn’t matter which one you substitute it into. I’ve just decided to substitute it into equation two.

So when I do that, I’m gonna get 5.5 minus 𝑦 is equal to two. So then to find 𝑦, the first thing I do is add 𝑦 to each side to make 𝑦 positive. So we get 5.5 equals two plus 𝑦. And then I’m gonna subtract two from each side of the equation. When we do that, we get 3.5 is equal to 𝑦. So therefore, we can say that the solution to the pair of equations three 𝑥 minus two 𝑦 equals 9.5 and 𝑥 minus 𝑦 equals two is 𝑥 is equal to 5.5 and 𝑦 is equal to 3.5. And if you wanted to check, you can substitute these into one of our other equations. And they would give us the correct answers.

Okay, so this was the first method. This was using elimination. I also said I’ll show you substitution. So to use substitution, we’ll again have them labeled one and two. So our equations are labeled one and two. But this time, to make equation three, what we’re going to do is rearrange one of our equations to make 𝑥 or 𝑦 the subject of the equation. So that’s why it’s a good method for this particular pair of equations because we can see that one of them can be rearranged easily to make 𝑥 or 𝑦 the subject.

So therefore, to create equation three, what we can do is add 𝑦 to each side of the equation in equation two, cause when we do that, we’re gonna get 𝑥 is equal to two plus 𝑦. So now what we do is we substitute 𝑥 equals two plus 𝑦 into equation one.

So we now can actually put two plus 𝑦 instead of 𝑥 in that equation. So when we do that, we can see that we no longer have any 𝑥s and we’re just dealing with 𝑦s. So we can now solve this equation to find 𝑦. So first of all, we expand the bracket. So we get three multiplied by two is six. Then three multiplied by positive 𝑦 gives us positive three 𝑦. Then we’ve got minus two 𝑦 equals 9.5.

So then what we’re gonna do is subtract six from each side of the equation. So we’re left with the 𝑦 terms on the left-hand side. And then we’re also gonna collect our 𝑦 terms. So we’ve got positive three 𝑦 minus two 𝑦, which is just gonna be 𝑦. And then we have 9.5 minus six, which is gonna be equal to 3.5. So we’ve got 𝑦 is equal to 3.5. And this agrees with our value for 𝑦 from our elimination method.

So now we can move on and find 𝑥. So now to find 𝑥, what we’re gonna do is substitute 𝑦 equals 3.5 into equation three. So when we do that, we get 𝑥 is equal to two plus 3.5. So we get 𝑥 is equal to 5.5. And this is again the same as the value we got with the elimination method.

So therefore, we’ve used both the elimination and substitution methods and shown that the solution to the pair of equations is 𝑥 equals 5.5 and 𝑦 equals 3.5.

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